We present a method for a certain class of Markov Decision Processes (MDPs) that can relate the optimal policy back to one or more reward sources in the environment. For a given initial state, without fully computing the value function, q-value function, or the optimal policy the algorithm can determine which rewards will and will not be collected, whether a given reward will be collected only once or continuously, and which local maximum within the value function the initial state will ultimately lead to. We demonstrate that the method can be used to map the state space to identify regions that are dominated by one reward source and can fully analyze the state space to explain all actions. We provide a mathematical framework to show how all of this is possible without first computing the optimal policy or value function.
Markov Decision Processes (MDPs) are a mathematical framework for modeling sequential decision making under uncertainty. The classical approaches for solving MDPs are well known and have been widely studied, some of which rely on approximation techniques to solve MDPs with large state space and/or action space. However, most of these classical solution approaches and their approximation techniques still take much computation time to converge and usually must be re-computed if the reward function is changed. This paper introduces a novel alternative approach for exactly and efficiently solving …
Meta AI；UMPA, ENS Lyon（里昂高等师范学院纯数学与应用数学实验室）；Univ. Lille（里尔大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Centrale Lille（里尔中央理工学院）
In probably approximately correct (PAC) reinforcement learning (RL), an agent is required to identify an $ε$-optimal policy with probability $1-δ$. While minimax optimal algorithms exist for this problem, its instance-dependent complexity remains elusive in episodic Markov decision processes (MDPs). In this paper, we propose the first nearly matching (up to a horizon squared factor and logarithmic terms) upper and lower bounds on the sample complexity of PAC RL in deterministic episodic MDPs with finite state and action spaces. In particular, our bounds feature a new notion of sub-optimality g…
We propose an algorithm for deterministic continuous Markov Decision Processes with sparse rewards that computes the optimal policy exactly with no dependency on the size of the state space. The algorithm has time complexity of $O( |R|^3 \times |A|^2 )$ and memory complexity of $O( |R| \times |A| )$, where $|R|$ is the number of reward sources and $|A|$ is the number of actions. Furthermore, we describe a companion algorithm that can follow the optimal policy from any initial state without computing the entire value function, instead computing on-demand the value of states as they are needed. …
We present new algorithms for computing and approximating bisimulation metrics in Markov Decision Processes (MDPs). Bisimulation metrics are an elegant formalism that capture behavioral equivalence between states and provide strong theoretical guarantees on differences in optimal behaviour. Unfortunately, their computation is expensive and requires a tabular representation of the states, which has thus far rendered them impractical for large problems. In this paper we present a new version of the metric that is tied to a behavior policy in an MDP, along with an analysis of its theoretical prop…
Inria（法国国家信息与自动化研究所）；Département d’Informatique de l’Ecole Normale Supérieure（巴黎高等师范学院计算机系）；PSL Research University（巴黎文理研究大学）
We consider deterministic Markov decision processes (MDPs) and apply max-plus algebra tools to approximate the value iteration algorithm by a smaller-dimensional iteration based on a representation on dictionaries of value functions. The setup naturally leads to novel theoretical results which are simply formulated due to the max-plus algebra structure. For example, when considering a fixed (non adaptive) finite basis, the computational complexity of approximating the optimal value function is not directly related to the number of states, but to notions of covering numbers of the state space. …
We consider Markov Decision Processes (MDPs) with deterministic transitions and study the problem of regret minimization, which is central to the analysis and design of optimal learning algorithms. We present logarithmic problem-specific regret lower bounds that explicitly depend on the system parameter (in contrast to previous minimax approaches) and thus, truly quantify the fundamental limit of performance achievable by any learning algorithm. Deterministic MDPs can be interpreted as graphs and analyzed in terms of their cycles, a fact which we leverage in order to identify a class of determ…
TTIC（丰田技术研究所）；Microsoft Research（微软研究院）；University of Michigan（密歇根大学）
We consider a Markov decision process with deterministic state transition dynamics, adversarially generated rewards that change arbitrarily from round to round, and a bandit feedback model in which the decision maker only observes the rewards it receives. In this setting, we present a novel and efficient online decision making algorithm named MarcoPolo. Under mild assumptions on the structure of the transition dynamics, we prove that MarcoPolo enjoys a regret of O(T^(3/4)sqrt(log(T))) against the best deterministic policy in hindsight. Specifically, our analysis does not rely on the stringent …
Current work in explainable reinforcement learning generally produces policies in the form of a decision tree over the state space. Such policies can be used for formal safety verification, agent behavior prediction, and manual inspection of important features. However, existing approaches fit a decision tree after training or use a custom learning procedure which is not compatible with new learning techniques, such as those which use neural networks. To address this limitation, we propose a novel Markov Decision Process (MDP) type for learning decision tree policies: Iterative Bounding MDPs (…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of Michigan（密歇根大学）
It is a remarkable fact that the same $O(\sqrt{T})$ regret rate can be achieved in both the Experts Problem and the Adversarial Multi-Armed Bandit problem albeit with a worse dependence on number of actions in the latter case. In contrast, it has been shown that handling online MDPs with communicating structure and bandit information incurs $Ω(T^{2/3})$ regret even in the case of deterministic transitions. Is this the price we pay for handling communicating structure or is it because we also have bandit feedback? In this paper we show that with full information, online MDPs can still be learne…
We propose a new approach to the problem of searching a space of policies for a Markov decision process (MDP) or a partially observable Markov decision process (POMDP), given a model. Our approach is based on the following observation: Any (PO)MDP can be transformed into an "equivalent" POMDP in which all state transitions (given the current state and action) are deterministic. This reduces the general problem of policy search to one in which we need only consider POMDPs with deterministic transitions. We give a natural way of estimating the value of all policies in these transformed POMDPs. P…
DeepMind；University College London（伦敦大学学院）；University of Illinois（伊利诺伊大学）；University of Cambridge（剑桥大学）；University of Alberta（阿尔伯塔大学）
We consider local planning in fixed-horizon MDPs with a generative model under the assumption that the optimal value function lies close to the span of a feature map. The generative model provides a local access to the MDP: The planner can ask for random transitions from previously returned states and arbitrary actions, and features are only accessible for states that are encountered in this process. As opposed to previous work (e.g. Lattimore et al. (2020)) where linear realizability of all policies was assumed, we consider the significantly relaxed assumption of a single linearly realizable …
We propose a distributionally robust return-risk model for Markov decision processes (MDPs) under risk and reward ambiguity. The proposed model optimizes the weighted average of mean and percentile performances, and it covers the distributionally robust MDPs and the distributionally robust chance-constrained MDPs (both under reward ambiguity) as special cases. By considering that the unknown reward distribution lies in a Wasserstein ambiguity set, we derive the tractable reformulation for our model. In particular, we show that that the return-risk model can also account for risk from uncertain…
University of Twente（特文特大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）；University of Liverpool（利物浦大学）
When omega-regular objectives were first proposed in model-free reinforcement learning (RL) for controlling MDPs, deterministic Rabin automata were used in an attempt to provide a direct translation from their transitions to scalar values. While these translations failed, it has turned out that it is possible to repair them by using good-for-MDPs (GFM) Büchi automata instead. These are nondeterministic Büchi automata with a restricted type of nondeterminism, albeit not as restricted as in good-for-games automata. Indeed, deterministic Rabin automata have a pretty straightforward translation to…
We study countably infinite Markov decision processes (MDPs) with real-valued transition rewards. Every infinite run induces the following sequences of payoffs: 1. Point payoff (the sequence of directly seen transition rewards), 2. Total payoff (the sequence of the sums of all rewards so far), and 3. Mean payoff. For each payoff type, the objective is to maximize the probability that the $\liminf$ is non-negative. We establish the complete picture of the strategy complexity of these objectives, i.e., how much memory is necessary and sufficient for $\varepsilon$-optimal (resp. optimal) strategi…
DeepMind（深度思维）；University College London（伦敦大学学院）；University of Alberta（阿尔伯塔大学）
We consider the minimax query complexity of online planning with a generative model in fixed-horizon Markov decision processes (MDPs) with linear function approximation. Following recent works, we consider broad classes of problems where either (i) the optimal value function $v^\star$ or (ii) the optimal action-value function $q^\star$ lie in the linear span of some features; or (iii) both $v^\star$ and $q^\star$ lie in the linear span when restricted to the states reachable from the starting state. Recently, Weisz et al. (2021b) showed that under (ii) the minimax query complexity of any plann…
Policy Iteration (PI) is a classical family of algorithms to compute an optimal policy for any given Markov Decision Problem (MDP). The basic idea in PI is to begin with some initial policy and to repeatedly update the policy to one from an improving set, until an optimal policy is reached. Different variants of PI result from the (switching) rule used for improvement. An important theoretical question is how many iterations a specified PI variant will take to terminate as a function of the number of states $n$ and the number of actions $k$ in the input MDP. While there has been considerable p…
Value iteration is a commonly used and empirically competitive method in solving many Markov decision process problems. However, it is known that value iteration has only pseudo-polynomial complexity in general. We establish a somewhat surprising polynomial bound for value iteration on deterministic Markov decision (DMDP) problems. We show that the basic value iteration procedure converges to the highest average reward cycle on a DMDP problem in heta(n^2) iterations, or heta(mn^2) total time, where n denotes the number of states, and m the number of edges. We give two extensions of value itera…
On the Convergence of Monte Carlo UCB for Random-Length Episodic MDPs
SFSC of AI and DL, NYU Shanghai（纽约大学上海人工智能与深度学习科学与技术中心）；New York University（纽约大学）；New York University Shanghai（纽约大学上海分校）；New York University Abu Dhabi（纽约大学阿布扎比分校）
In this work, we provide theoretical guarantees for reward decomposition in deterministic MDPs. Reward decomposition is a special case of Hierarchical Reinforcement Learning, that allows one to learn many policies in parallel and combine them into a composite solution. Our approach builds on mapping this problem into a Reward Discounted Traveling Salesman Problem, and then deriving approximate solutions for it. In particular, we focus on approximate solutions that are local, i.e., solutions that only observe information about the current state. Local policies are easy to implement and do not r…
DeepMind（深度思维）；University College London（伦敦大学学院）；University of Alberta（阿尔伯塔大学）；Omron Sinic X（欧姆龙Sinic X）
We consider approximate dynamic programming in $γ$-discounted Markov decision processes and apply it to approximate planning with linear value-function approximation. Our first contribution is a new variant of Approximate Policy Iteration (API), called Confident Approximate Policy Iteration (CAPI), which computes a deterministic stationary policy with an optimal error bound scaling linearly with the product of the effective horizon $H$ and the worst-case approximation error $ε$ of the action-value functions of stationary policies. This improvement over API (whose error scales with $H^2$) comes…
University of Waterloo（滑铁卢大学）；Monash University（莫纳什大学）
A promising characteristic of Deep Reinforcement Learning (DRL) is its capability to learn optimal policy in an end-to-end manner without relying on feature engineering. However, most approaches assume a fully observable state space, i.e. fully observable Markov Decision Processes (MDPs). In real-world robotics, this assumption is unpractical, because of issues such as sensor sensitivity limitations and sensor noise, and the lack of knowledge about whether the observation design is complete or not. These scenarios lead to Partially Observable MDPs (POMDPs). In this paper, we propose Long-Short…
UvA-Bosch Deltalab（阿姆斯特丹大学-博世三角洲实验室）；University of Amsterdam（阿姆斯特丹大学）；Delft University of Technology（代尔夫特理工大学）；Google Research, Brain Team（谷歌研究院大脑团队）
This work exploits action equivariance for representation learning in reinforcement learning. Equivariance under actions states that transitions in the input space are mirrored by equivalent transitions in latent space, while the map and transition functions should also commute. We introduce a contrastive loss function that enforces action equivariance on the learned representations. We prove that when our loss is zero, we have a homomorphism of a deterministic Markov Decision Process (MDP). Learning equivariant maps leads to structured latent spaces, allowing us to build a model on which we p…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Google Research（谷歌研究院）；The Hong Kong University of Science and Technology（香港科技大学）
Recently, several studies (Zhou et al., 2021a; Zhang et al., 2021b; Kim et al., 2021; Zhou and Gu, 2022) have provided variance-dependent regret bounds for linear contextual bandits, which interpolates the regret for the worst-case regime and the deterministic reward regime. However, these algorithms are either computationally intractable or unable to handle unknown variance of the noise. In this paper, we present a novel solution to this open problem by proposing the first computationally efficient algorithm for linear bandits with heteroscedastic noise. Our algorithm is adaptive to the unkno…
Data61/CSIRO（澳大利亚联邦科学与工业研究组织数据61）；Research School of Computer Science, ANU（澳大利亚国立大学计算机科学研究学院）
Markov decision processes (MDPs) are the standard formalism for modelling sequential decision making in stochastic environments. Policy synthesis addresses the problem of how to control or limit the decisions an agent makes so that a given specification is met. In this paper we consider PCTL*, the probabilistic counterpart of CTL*, as the specification language. Because in general the policy synthesis problem for PCTL* is undecidable, we restrict to policies whose execution history memory is finitely bounded a priori. Surprisingly, no algorithm for policy synthesis for this natural and express…
In the optimization of dynamical systems, the variables typically have constraints. Such problems can be modeled as a constrained Markov Decision Process (CMDP). This paper considers a model-free approach to the problem, where the transition probabilities are not known. In the presence of long-term (or average) constraints, the agent has to choose a policy that maximizes the long-term average reward as well as satisfy the average constraints in each episode. The key challenge with the long-term constraints is that the optimal policy is not deterministic in general, and thus standard Q-learning…
The OpenAI Gym provides researchers and enthusiasts with simple to use environments for reinforcement learning. Even the simplest environment have a level of complexity that can obfuscate the inner workings of RL approaches and make debugging difficult. This whitepaper describes a Python framework that makes it very easy to create simple Markov-Decision-Process environments programmatically by specifying state transitions and rewards of deterministic and non-deterministic MDPs in a domain-specific language in Python. It then presents results and visualizations created with this MDP framework.
Arizona State University（亚利桑那州立大学）；United Technologies Research Center（联合技术研究中心）；University of Massachusetts（马萨诸塞大学）
Integrated task and motion planning has emerged as a challenging problem in sequential decision making, where a robot needs to compute high-level strategy and low-level motion plans for solving complex tasks. While high-level strategies require decision making over longer time-horizons and scales, their feasibility depends on low-level constraints based upon the geometries and continuous dynamics of the environment. The hybrid nature of this problem makes it difficult to scale; most existing approaches focus on deterministic, fully observable scenarios. We present a new approach where the high…
Google Research（谷歌研究院）；Johns Hopkins University（约翰斯·霍普金斯大学）；Courant Institute（库朗研究所）
We provide improved gap-dependent regret bounds for reinforcement learning in finite episodic Markov decision processes. Compared to prior work, our bounds depend on alternative definitions of gaps. These definitions are based on the insight that, in order to achieve a favorable regret, an algorithm does not need to learn how to behave optimally in states that are not reached by an optimal policy. We prove tighter upper regret bounds for optimistic algorithms and accompany them with new information-theoretic lower bounds for a large class of MDPs. Our results show that optimistic algorithms ca…
New York University Shanghai（上海纽约大学）；New York University（纽约大学）
A simple and natural algorithm for reinforcement learning (RL) is Monte Carlo Exploring Starts (MCES), where the Q-function is estimated by averaging the Monte Carlo returns, and the policy is improved by choosing actions that maximize the current estimate of the Q-function. Exploration is performed by "exploring starts", that is, each episode begins with a randomly chosen state and action, and then follows the current policy to the terminal state. In the classic book on RL by Sutton & Barto (2018), it is stated that establishing convergence for the MCES algorithm is one of the most important …
UC Berkeley（加州大学伯克利分校）；EPFL（洛桑联邦理工学院）；UC San Diego（加州大学圣迭戈分校）；UC Los Angeles（加州大学洛杉矶分校）
Planning in adversarial and uncertain environments can be modeled as the problem of devising strategies in stochastic perfect information games. These games are generalizations of Markov decision processes (MDPs): there are two (adversarial) players, and a source of randomness. The main practical obstacle to computing winning strategies in such games is the size of the state space. In practice therefore, one typically works with abstractions of the model. The diffculty is to come up with an abstraction that is neither too coarse to remove all winning strategies (plans), nor too fine to be intr…
University of Twente（特文特大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）；University of Liverpool（利物浦大学）
Recursion is the fundamental paradigm to finitely describe potentially infinite objects. As state-of-the-art reinforcement learning (RL) algorithms cannot directly reason about recursion, they must rely on the practitioner's ingenuity in designing a suitable "flat" representation of the environment. The resulting manual feature constructions and approximations are cumbersome and error-prone; their lack of transparency hampers scalability. To overcome these challenges, we develop RL algorithms capable of computing optimal policies in environments described as a collection of Markov decision pro…
Deep reinforcement learning (DRL) on Markov decision processes (MDPs) with continuous action spaces is often approached by directly training parametric policies along the direction of estimated policy gradients (PGs). Previous research revealed that the performance of these PG algorithms depends heavily on the bias-variance tradeoffs involved in estimating and using PGs. A notable approach towards balancing this tradeoff is to merge both on-policy and off-policy gradient estimations. However existing PG merging methods can be sample inefficient and are not suitable to train deterministic polic…
Reinforcement learning with function approximation has recently achieved tremendous results in applications with large state spaces. This empirical success has motivated a growing body of theoretical work proposing necessary and sufficient conditions under which efficient reinforcement learning is possible. From this line of work, a remarkably simple minimal sufficient condition has emerged for sample efficient reinforcement learning: MDPs with optimal value function $V^*$ and $Q^*$ linear in some known low-dimensional features. In this setting, recent works have designed sample efficient algo…
We consider a settings of hierarchical reinforcement learning, in which the reward is a sum of components. For each component we are given a policy that maximizes it and our goal is to assemble a policy from the individual policies that maximizes the sum of the components. We provide theoretical guarantees for assembling such policies in deterministic MDPs with collectible rewards. Our approach builds on formulating this problem as a traveling salesman problem with discounted reward. We focus on local solutions, i.e., policies that only use information from the current state; thus, they are ea…
